The motivation

So, we have a sample of data

We assume that these data come from some known distribution with certain parameters

And we want to find our “best guess” for these parameters

Can we use our data sample to help us?

Here, the sample average seems a sensible guess for the population mean

What about \(\sigma\)? Is 1 a sensible guess?

Is \(\mu = 10\) our “best” possible guess assuming \(\sigma = 1\)

Is \(\sigma = 1\) our “best” possible guess assuming \(\mu = 10\)

What we actually want is to “guess” both together.